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Definitions in science

How things are defined is very important in all aspects of life. Laws, for example, need clear definitions so that they can be applied appropriately. In science the same is true, definitions allow us to be on the same page when we are describing phenomena; they give us a common language. Or least that’s the idea. Often people define things in different ways. Often people make assumptions about how things are defined based on past experience, and where clear definitions are not given this can lead to confusion and mistakes.

A simple example can be seen in some work I have been doing this week to look at optical interactions with metallic and semiconductor materials. In one text book the complex refractive index (which takes into account the conventional refractive index and the absoprtion of the material) was written like this:

\hat{n} = n -i\kappa

where \hat{n} is the complex refractive index, n is the refractive index and \kappa is the material absorption. Now normally this would appear with a ‘plus’ sign rather than a ‘minus’, but I initially just assumed that this meant that the authors were defining the absorption coefficient as a negative, rather than the usual positive (\kappa >0 normally implies an absoprtion loss). Needing some more info on some of the derivations in the textbook I got out another and there the equation appears in it’s more conventional form – but it soon became clear that the original definition was not using a negative kappa, and that by missing out exactly how the parameters were defined the original text had led me a merry dance. This sort of stuff is not unusual, and can be a bit annoying.

The reason for the confusion between the two definitions is related to how complex waves are defined, and there is more on this in the wikipedia articles on opacity and refractive index.

In this case, no great harm was done, other than waste a bit of my time. However in other cases it can lead to the misinterpretation of experimental results. As a PhD student I worked on atomic and laser physics and you have to deal with a number of ‘frequency’ terms, such as state lifetimes and interaction strengths, which can be described by a ‘Rabi frequency‘. At the start of my PhD the group I worked in had been doing very successful experiments on atomic rubidium and sodium for a number of years. They had developed models which successfully, but qualitatively (it turns out) described what they were seeing. When one of the group started to look at how the models could be used to predict laser gain on some of the transitions we were looking at, the numbers didn’t make sense. So we started to look at how the equations were all derived and where all the numbers (mostly atomic lifetimes, but also Rabi frequencies) came from. We started to look at a wide range of literature that discussed the values of these lifetimes and we found that there were two sets of inconsistencies. The first related to how a lifetime is defined in terms of the decay process. This decay gives rise to a ‘linewidth’ of a transition, and this linewidth can be defined, as can most distributions as the width at full width half maximum (FWHM), half width half maximum (HWHM) etc. This we found wasn’t two bad. We had a handle on this, and people tended to be careful.

What people were less careful with was the units they were using. As I explained to the Advanced Higher Physics students at our rotational motion workshop last week, rotational quantities have very similar forms to their linear counterparts, but they have different units. Frequency, for example, is measured in Hertz (which is just ‘per second’):

f =\frac{1}{T} (Hz)

but rotational frequency also includes angle. Most students include the ‘radians’ when dealing with such quantities. I have always been taught that the ‘radians’ is superfluous, as it’s not really a ‘proper’ unit. So angular frequency has the units ‘per second’:

\omega=\frac{2\pi}{T} (s^{-1})

In the papers we looked at, people were very sloppy (and often incorrect) in how they applied these units – they apply to the lifetimes of the atomic states, just they can be applied to a particle on a string spinning round my head. So it turned out that we were missing a factor of 2\pi in all our equations. As these where being integrated over a range of frequencies, this turned out to have a huge effect on the calculations (it slowed them down massively, for a start), but it fixed our initial problem, and my colleague was then able to make quantitative predictions using the new results.

As someone once told my supervisor about another such incident, “…they missed out 2pi, and let me tell you, 2pi is a big number!” Indeed.

The moral of this tale, is that definitions are really, really important. If you are not clear about how something is defined, particularly when there is more than one definition, you are opening up your work to misinterpretation, and probably will inflict many wasted hours on some poor unsuspecting graduate student, and ultimately you can effect the predictions that others can make with your work. This might seem a slight point, but when you are using the calculations to design an experiment – say the prediction is you must use a 4W 1064nm laser as opposed to a 4W 532nm, but it’s really vice versa – it can translate into money as much as time.

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